## Definition

Valuation of financial instruments and project valuation techniques usually assume that expected cash flows are discounted at discrete intervals, e.g., daily, monthly, quarterly, semiannually, or annually. In some instances, however, especially for high-risk investments, continuous discounting can be used for more precise valuation. For example, the technique of continuous discounting is widely used in financial option valuation and namely in the Black-Scholes option pricing model.

## Formula

To calculate the present value of a cash flow, use the following formula of continuous discounting.

Here FV is the future value, r is the annual interest rate, t is the number of years, and e is Euler’s number equal to 2.71828.

## Example

An individual has the possibility of investing $20,000 and getting back a lump sum of $30,000 after 5 years. It is necessary to decide whether or not to accept or reject this investment opportunity if the required rate of return on investments with the same degree of risk is 8% (annually) and interest is continuously compounded.

To solve the problem, we need to use the equation above.

The present value of cash flow is $20,109.60, which is more than $20,000, so the investment should be accepted.

## Continuous Discounting vs. Discrete Discounting

The difference between discrete and continuous discounting is shown in the figure below.

Let’s assume what the present value of $1 should be if it is discounted at an annual discount rate of 15% annually (discretely) and continuously. For example, if we expect $1 to be received at the end of the first year, its present value is $0.8696 at annual discounting and $0.8607 at continuous discounting.

PV = $1 ÷ (1 + 0.15)^{1} = $0.8696

PV = $1 ÷ 2.71828^{0.15×1} = $0.8607

The present value of $1 to be received at the end of the fifth year is $0.4972 and $0.4724 respectively.

PV = $1 ÷ (1 + 0.15)^{5} = $0.4972

PV = $1 ÷ 2.71828^{0.15×5} = $0.4724

The present value of $1 to be received at the end of tenth year is $0.2472 and $0.2231 respectively.

PV = $1 ÷ (1 + 0.15)^{10} = $0.2472

PV = $1 ÷ 2.71828^{0.15×10} = $0.2231

The present value of $1 to be received at the end of the twentieth year is $0.0611 and $0.0498 respectively.

PV = $1 ÷ (1 + 0.15)^{20} = $0.0611

PV = $1 ÷ 2.71828^{0.15×20} = $0.0498

As we can see, continuous discounting always leads to a lower present value than discrete discounting. In the early years, the difference between values increases, but the more the future due date is extended, the lower the difference in present values will be.